文摘
Following Schachermayer, a subset B of an algebra A of subsets of Ω is said to have the N-property if a B-pointwise bounded subset M of ba(A) is uniformly bounded on A, where ba(A) is the Banach space of the real (or complex) finitely additive measures of bounded variation defined on A. Moreover B is said to have the strong N-property if for each increasing countable covering (Bm)m of B there exists Bn which has the N-property. The classical Nikodym–Grothendieck's theorem says that each σ -algebra S of subsets of Ω has the N-property. The Valdivia's theorem stating that each σ -algebra S has the strong N -property motivated the main measure-theoretic result of this paper: We show that if (Bm1)m1 is an increasing countable covering of a σ -algebra S and if (Bm1,m2,…,mp,mp+1)mp+1 is an increasing countable covering of Bm1,m2,…,mp, for each p,mi∈N, 1⩽i⩽p, then there exists a sequence (ni)i such that each Bn1,n2,…,nr, r∈N, has the strong N -property. In particular, for each increasing countable covering (Bm)m of a σ -algebra S there exists Bn which has the strong N-property, improving mentioned Valdivia's theorem. Some applications to localization of bounded additive vector measures are provided.